Finite Local Information Capacity in Continuum Dynamics

Finite-dimensional local Hilbert spaces are widely used as microscopic regulators and have also been proposed as a fundamental information-capacity property of space-time. A finite local dimension, however, is not equivalent to a spatial lattice spacing, and neither is equivalent to a coarse-graining scale. We formulate a regulator-resolved framework that keeps these three notions separate and defines finite-information deviations operationally as the failure of microscopic and effective time evolution to commute with a specified coarse-graining channel. The resulting deviation is a state- and observable-dependent quantity in a common effective state space rather than a formal subtraction of generators acting on different domains. Standard trace-distance, truncation, entropy-continuity, and multi-information results are assembled into a matching-path upper bound resolving capacity, spatial-discretization, and coarse-graining contributions; a finite-time capacity bound controlled by discarded state weight and leakage through the cutoff boundary; observable error bounds; and an exact distinction between block entropy and averaged one-cell entropy. An information-geometric expansion of a chosen nearest-neighbor distinguishability functional produces a leading Euclidean continuum form of nonlinear-sigma-model type. It is not an effective action derived from the microscopic Hamiltonian; a Lorentzian dynamical theory, and in particular a single entropy field, requires additional continuation, closure, and matching assumptions. Finally, we test finite-capacity convergence in an exactly evolved four-site periodic Klein–Gordon chain at the single lattice spacing a=1, varying the local number-state cutoff. Spectral peaks are extracted from dynamical matrix elements rather than imposed, while the dependence on a is evaluated analytically from the exact lattice dispersion. Ground-state, reduced-state, and entropy errors converge rapidly with local capacity, whereas the lattice–continuum discrepancy remains at fixed spacing. The contribution is a regulator-resolved conceptual framework and a reproducible benchmark for finite-information models, without claiming a new inequality or a unique microscopic theory of quantum gravity.